Chapter One: Vector Analysis The use of vectors and vector analysis ...

Electromagnetic Theorem
(Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department
Problem:
Let 2 3 ,
3 2
Check product rules:
1‐ ∙ ∙ ∙
2‐ ∙ ∙ ∙
3‐ ∙ ∙ ∙ ∙
Proof of (1):
2 36 9 2 6
3 2 0
∙ 6 9 2 6
∙ 6 9 0 15
3 2 3 2 0
∴B ∙ 0
0 2 3 0 2 3
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∴A ∙ 15
∴ ∙ ∙ 015 15
∴ ∙ ∙ ∙
Second Derivatives:
The gradient, the divergence, and curl are the only first derivative we can make with ; by
applying twice we can construct five species of second derivatives,
The gradient is a vector, so we can take the divergence and curl of it:
1‐ Divergence of gradient: ∙
2‐ Curl of gradient:
The divergence ∙ is a scalar, all we can do is take its gradient:
3‐ Gradient of divergence: ∙
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